| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Integrate after simplifying a quotient |
| Difficulty | Moderate -0.3 This is a multi-part question covering standard A-level techniques: (a)(i) requires simple algebraic manipulation before integration, (a)(ii) uses the standard cos²x identity, (b) is routine trapezium rule application, and (c) involves the R-cos(θ-α) form which is a standard Pure 2 topic. All parts are textbook exercises requiring recall and direct application of learned methods with no novel problem-solving, making it slightly easier than average. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals1.06d Natural logarithm: ln(x) function and properties1.08b Integrate x^n: where n != -1 and sums1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) Attempt to divide by \(e^{2x}\) and attempt to integrate 2 terms | M1 | |
| Integrate a term of form \(ke^{-2x}\) correctly | A1\(\checkmark\) | |
| Fully correct integral \(x - 3e^{-2x} (+c)\) | A1 | [3] |
| (ii) State correct expression \(\frac{1}{4}\cos 2x + \frac{1}{2}\) or equivalent | B1 | |
| Integrate an expression of the form \(a + b\cos 2x\), where \(ab \neq 0\), correctly | M1 | |
| State correct integral \(\frac{3\sin 2x}{4} + \frac{3x}{2} (+c)\) | A1 | [3] |
| (b) State or imply correct ordinates \(5.46143\ldots, 4.78941\ldots, 4.32808\ldots\) | B1 | |
| Use correct formula, or equivalent, correctly with \(h = 0.5\) and three ordinates | M1 | |
| Obtain answer 4.84 with no errors seen | A1 | [3] |
**(a) (i)** Attempt to divide by $e^{2x}$ and attempt to integrate 2 terms | M1 |
Integrate a term of form $ke^{-2x}$ correctly | A1$\checkmark$ |
Fully correct integral $x - 3e^{-2x} (+c)$ | A1 | [3]
**(ii)** State correct expression $\frac{1}{4}\cos 2x + \frac{1}{2}$ or equivalent | B1 |
Integrate an expression of the form $a + b\cos 2x$, where $ab \neq 0$, correctly | M1 |
State correct integral $\frac{3\sin 2x}{4} + \frac{3x}{2} (+c)$ | A1 | [3]
**(b)** State or imply correct ordinates $5.46143\ldots, 4.78941\ldots, 4.32808\ldots$ | B1 |
Use correct formula, or equivalent, correctly with $h = 0.5$ and three ordinates | M1 |
Obtain answer 4.84 with no errors seen | A1 | [3]
---6
\begin{enumerate}[label=(\alph*)]
\item Find
\begin{enumerate}[label=(\roman*)]
\item $\int \frac { \mathrm { e } ^ { 2 x } + 6 } { \mathrm { e } ^ { 2 x } } \mathrm {~d} x$,
\item $\int 3 \cos ^ { 2 } x \mathrm {~d} x$.
\end{enumerate}\item Use the trapezium rule with 2 intervals to estimate the value of
$$\int _ { 1 } ^ { 2 } \frac { 6 } { \ln ( x + 2 ) } \mathrm { d } x$$
giving your answer correct to 2 decimal places.
\begin{enumerate}[label=(\roman*)]
\item Express $3 \cos \theta + \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$, giving the exact value of $R$ and the value of $\alpha$ correct to 2 decimal places.
\item Hence solve the equation
$$3 \cos 2 x + \sin 2 x = 2$$
giving all solutions in the interval $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2013 Q6 [9]}}